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Find integration of :- cosxlog(tanx2)dx=
(a) sinxlog|tanx|x+C
(b) sinxlog|tanx2|+x+C
(c) sinxlog|tanx2|x+C
(d) sinxlog|tanx2|x+C

Answer
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Hint: Here, we need to find the value of the given integral. We will use integration by parts and trigonometric identities to simplify the given equation and find the integral of the given function. Integration is a method by which we can find summation of discrete data.

Formula Used: We will use the following formula:
1.Using integration by parts, the integral of the product of two differentiable functions of x can be written as uvdx=uvdx(d(u)dx×vdx)dx, where u and v are the differentiable functions of x.
2.The tangent of an angle θ can be written as tanθ=sinθcosθ.
3.The secant of an angle θ can be written as secθ=1cosθ.

Complete step-by-step answer:
We will integrate the simplified function using integration by parts.
Rewriting the equation, we get
cosxlog(tanx2)dx=cosx×log(tanx2)dx
Let u be log(tanx2) and v be cosx.
Therefore, by integrating cosx×log(tanx2)dx by parts, we get
cosx×log(tanx2)dx=log(tanx2)(cosx)dx[d{log(tanx2)}dx×(cosx)dx]dx
The derivative of a function of the form log{f(x)} is 1f(x)×d{f(x)}dx.
Thus, we get
cosx×log(tanx2)dx=log(tanx2)(cosx)dx[1tanx2×d(tanx2)dx×(cosx)dx]dx
Simplifying the derivative, we get
cosx×log(tanx2)dx=log(tanx2)(cosx)dx[1tanx2×sec2x2×d(x2)dx×(cosx)dx]dx
Therefore, we get
cosx×log(tanx2)dx=log(tanx2)(cosx)dx[1tanx2×sec2x2×12×(cosx)dx]dx
The tangent of an angle θ can be written as tanθ=sinθcosθ.
The secant of an angle θ can be written as secθ=1cosθ.
Rewriting the terms in the parentheses in terms of sine and cosine, we get
cosx×log(tanx2)dx=log(tanx2)(cosx)dx[1sinx2cosx2×1cos2x2×12×(cosx)dx]dx
Simplifying the expression, we get
cosx×log(tanx2)dx=log(tanx2)(cosx)dx[cosx2sinx2×1cos2x2×12×(cosx)dx]dxcosx×log(tanx2)dx=log(tanx2)(cosx)dx[12×1sinx2×1cosx2×(cosx)dx]dx
Therefore, we get
cosx×log(tanx2)dx=log(tanx2)(cosx)dx[12sinx2cosx2×(cosx)dx]dx
Applying the trigonometric identity 2sinAcosA=sin2A, we get
cosx×log(tanx2)dx=log(tanx2)(cosx)dx[1sinx×(cosx)dx]dx
Now, the integral of cosx is sinx.
Substituting cosxdx=sinx in the equation, we get
cosx×log(tanx2)dx=log(tanx2)sinx[1sinx×sinx]dx
Simplifying the expression, we get
cosx×log(tanx2)dx=sinxlog(tanx2)(1)dx
The integral of a constant with respect to a variable is the product of the constant and the variable.
Therefore, we get
cosxlog(tanx2)dx=sinxlog(tanx2)x+C, where C is the constant of integration
Therefore, we get the value of the integral cosxlog(tanx2)dx as sinxlog(tanx2)x+C, where C is the constant of integration.
Thus, the correct option is option (d).

Note: We simplified d(tanx2)dx as sec2x2×d(x2)dx. This is because the derivative of a function of the form tan{f(x)} is sec2{f(x)}×d{f(x)}dx.
We simplified d(x2)dx as 12. This is because the derivative of a function of the form af(x) is ad{f(x)}dx, and the derivative of a variable with respect to itself is 1.
We used the trigonometric identity 2sinAcosA=sin2A. This is the trigonometric identity for the sine of the double of an angle.